Using the van der Corput Lemma (see for example [1]}), composing with \(T^{-n}\) , and then using the Cauchy-Schwarz inequality, we get that it suffices to show that for every \(m\in {\mathbb {N}}\)  we have
\(\lim _{N\rightarrow \infty } {\mathbb {E}}_{n\in [N]} \, T^{(n+m)^2-n}f_2\cdot T^{n^2-n}\overline{f}_2= 0.\) 
Using the van der Corput lemma again, composing with \(T^{-n^2}\) , and then using the Cauchy-Schwarz inequality, we further reduce matters to showing the following: If \(\vert \!|\!| f_2|\!|\!\vert _{3}=0\) , then for all \(g,h\in L^\infty (\mu )\)  and all \(a,b,c\in {\mathbb {N}}\)  with \(a\ne b, a\ne c\) , we have
\(\lim _{N\rightarrow \infty } {\mathbb {E}}_{n\in [N]} \, T^{an}f_2\cdot T^{bn}g\cdot T^{cn}h= 0.\) 
Since \(\vert \!|\!| f_2|\!|\!\vert _3=0\) , this follows from [2]}.The argument in [2]} is non-trivial. One can avoid it by using that \(\vert \!|\!| f_2|\!|\!\vert _4=0\)  implies the identity (REF ) (see for example [4]}). The drawback is that the use of the 4-th seminorm would complicate our subsequent arguments; but still a proof that avoids deep machinery can be given by using the argument in Section .
